Calculus — Derivatives⏱ 6 min read

Chain Rule

dy/dx = (dy/du) · (du/dx)

What is the Chain Rule?

The chain rule differentiates composite functions — functions inside other functions. If y depends on u, and u in turn depends on x, the chain rule links all three derivatives together: differentiate the outer function first (treating the inner function as a single block), then multiply by the derivative of that inner function.

To recognize when the chain rule is needed, ask: "is there a function inside another function?" sin(3x), (x²+1)⁶, and e^(x²) are all composite — each is one function applied to the output of another.

What Each Variable Means

Outer function's outputy depends on u, which itself depends on x.

Inner functionThe function nested inside the outer function.

dy/du

Outer derivativeHow y changes with respect to u.

du/dx

Inner derivativeHow u changes with respect to x.

When to Use It

Differentiating any composite function — a function nested inside another
Combined with the product and quotient rules for more complex expressions
Anywhere a rate of change depends on an intermediate variable, like related-rates problems

Step-by-Step Examples

Example 1: A power of a linear expression

Problem: Differentiate y = (3x + 2)⁵

Identify outer and inner functions

Outer: u⁵ → 5u⁴. Inner: u = 3x + 2 → 3.

d/du[u⁵] = 5u⁴, du/dx = 3

Multiply the outer derivative by the inner derivative

Substitute u back in and multiply.

dy/dx = 5(3x+2)⁴ × 3

✓

Answer: dy/dx = 15(3x+2)⁴

Example 2: A trig function of a power

Problem: Differentiate y = sin(x²)

Identify outer and inner functions

Outer: sin(u) → cos(u). Inner: u = x² → 2x.

d/du[sin u] = cos u, du/dx = 2x

Multiply the two derivatives

Substitute u back in.

dy/dx = cos(x²) × 2x

✓

Answer: dy/dx = 2x·cos(x²)

Common Mistakes

Mistake: Differentiating only the outer function and forgetting to multiply by the inner function's derivative.
Fix: d/dx[(3x+2)⁵] is not 5(3x+2)⁴ alone — you must also multiply by du/dx = 3, giving 15(3x+2)⁴.
Mistake: Missing that a function is composite in the first place.
Fix: Ask whether there's a function inside another function. If the argument of sin, cos, a power, or e isn't just "x" by itself, the chain rule is needed.